$$\mathrm{evaluate} \\$$$$\underset{−\infty} {\overset{+\infty} {\int}}\frac{\mathrm{sin}\:{x}}{{x}}{dx} \\$$
$$\mathrm{Let}\:\mathrm{us}\:\mathrm{consider}\:\underset{\mathrm{0}} {\overset{\infty} {\int}}\:\underset{\mathrm{0}} {\overset{\infty} {\int}}\:{e}^{−{st}} \mathrm{sin}\:{t}\:{dt}\:{ds} \\$$$$\underset{\mathrm{0}} {\overset{\infty} {\int}}\:\underset{\mathrm{0}} {\overset{\infty} {\int}}\:{e}^{−{st}} \mathrm{sin}\:{t}\:{dt}\:{ds}=\underset{\mathrm{0}} {\overset{\infty} {\int}}\:\frac{\mathrm{1}}{\mathrm{1}+{s}^{\mathrm{2}} }{ds}=\left[\mathrm{arctan}\:{s}\right]_{\mathrm{0}} ^{\infty} =\frac{\pi}{\mathrm{2}}…\left(\mathrm{1}\right) \\$$$$\underset{\mathrm{0}} {\overset{\infty} {\int}}\:\underset{\mathrm{0}} {\overset{\infty} {\int}}\:{e}^{−{st}} \mathrm{sin}\:{t}\:{dt}\:{ds}=\underset{\mathrm{0}} {\overset{\infty} {\int}}\:\underset{\mathrm{0}} {\overset{\infty} {\int}}\:{e}^{−{st}} \mathrm{sin}\:{t}\:{ds}\:{dt} \\$$$$=\underset{\mathrm{0}} {\overset{\infty} {\int}}\:\frac{\mathrm{sin}\:{t}}{{t}}\:{dt}….\left(\mathrm{2}\right) \\$$$$\mathrm{From}\:\left(\mathrm{1}\right)\:\mathrm{and}\:\left(\mathrm{2}\right) \\$$$$\underset{\mathrm{0}} {\overset{\infty} {\int}}\:\frac{\mathrm{sin}\:{t}}{{t}}\:{dt}\:=\frac{\pi}{\mathrm{2}} \\$$$$\underset{−\infty} {\overset{\infty} {\int}}\:\frac{\mathrm{sin}\:{t}}{{t}}\:{dt}=\mathrm{2}\underset{\mathrm{0}} {\overset{\infty} {\int}}\:\frac{\mathrm{sin}\:{t}}{{t}}\:{dt}=\pi \\$$